Quantum Two

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Time Independent Approximation Methods

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As we have seen, the task of predicting the evolution of an isolated quantum mechanical system can be reduced to the solution of an appropriate eigenvalue equation involving the Hamiltonian of the system.

Unfortunately, only a small number of quantum mechanical systems are amenable to an exact solution.

Moreover, even when an exact solution to the eigenvalue problem is available, it is often useful to understand the behavior of the system in the presence of weak external fields that my be imposed in order to probe the structure of its stationary states.

In these situations approximate methods are required for calculating the eigenstates of the Hamiltonian in the presence of a perturbation that makes it difficult or impossible to obtain an exact solution.

4

As we have seen, the task of predicting the evolution of an isolated quantum mechanical system can be reduced to the solution of an appropriate eigenvalue equation involving the Hamiltonian of the system.

Unfortunately, only a small number of quantum mechanical systems are amenable to an exact solution.

Moreover, even when an exact solution to the eigenvalue problem is available, it is often useful to understand the behavior of the system in the presence of weak external fields that my be imposed in order to probe the structure of its stationary states.

In these situations approximate methods are required for calculating the eigenstates of the Hamiltonian in the presence of a perturbation that makes it difficult or impossible to obtain an exact solution.

5

As we have seen, the task of predicting the evolution of an isolated quantum mechanical system can be reduced to the solution of an appropriate eigenvalue equation involving the Hamiltonian of the system.

Unfortunately, only a small number of quantum mechanical systems are amenable to an exact solution.

Moreover, even when an exact solution to the eigenvalue problem is available, it is often useful to understand the behavior of the system in the presence of weak external fields that my be imposed in order to probe the structure of its stationary states.

In these situations approximate methods are required for calculating the eigenstates of the Hamiltonian in the presence of a perturbation that makes it difficult or impossible to obtain an exact solution.

6

As we have seen, the task of predicting the evolution of an isolated quantum mechanical system can be reduced to the solution of an appropriate eigenvalue equation involving the Hamiltonian of the system.

Unfortunately, only a small number of quantum mechanical systems are amenable to an exact solution.

Moreover, even when an exact solution to the eigenvalue problem is available, it is often useful to understand the behavior of the system in the presence of weak external fields that my be imposed in order to probe the structure of its stationary states.

In these situations approximate methods are required for calculating the eigenstates of the Hamiltonian in the presence of a perturbation that makes it difficult or impossible to obtain an exact solution.

7

There are two general approaches commonly taken in problems of this sort.

The first, referred to as the variational method, is most commonly used to obtain information about the ground state, and low lying excited states of the system.

The second, more systematic approach is generally referred to as time-independent perturbation theory, of which there are two versions: non-degenerate perturbation theory and degenerate perturbation theory.

Either of these approaches is generally applicable to any set of discrete levels and is not necessarily restricted to the solution of the energy eigenvalue problem, but can be applied to any observable with a discrete spectrum.

We begin with a study of the variational method, which is based upon two variation theorems, one fairly weak, and the other of which makes a stronger and more useful statement. 8

There are two general approaches commonly taken in problems of this sort.

The first, referred to as the variational method, is most commonly used to obtain information about the ground state, and low lying excited states of the system.

The second, more systematic approach is generally referred to as time-independent perturbation theory, of which there are two versions: non-degenerate perturbation theory and degenerate perturbation theory.

Either of these approaches is generally applicable to any set of discrete levels and is not necessarily restricted to the solution of the energy eigenvalue problem, but can be applied to any observable with a discrete spectrum.

We begin with a study of the variational method, which is based upon two variation theorems, one fairly weak, and the other of which makes a stronger and more useful statement. 9

There are two general approaches commonly taken in problems of this sort.

The first, referred to as the variational method, is most commonly used to obtain information about the ground state, and low lying excited states of the system.

The second, more systematic approach is generally referred to as time-independent perturbation theory, of which there are two versions: non-degenerate perturbation theory and degenerate perturbation theory.

Either of these approaches is generally applicable to any set of discrete levels and is not necessarily restricted to the solution of the energy eigenvalue problem, but can be applied to any observable with a discrete spectrum.

We begin with a study of the variational method, which is based upon two variation theorems, one fairly weak, and the other of which makes a stronger and more useful statement. 10

There are two general approaches commonly taken in problems of this sort.

The first, referred to as the variational method, is most commonly used to obtain information about the ground state, and low lying excited states of the system.

The second, more systematic approach is generally referred to as time-independent perturbation theory, of which there are two versions: non-degenerate perturbation theory and degenerate perturbation theory.

Either of these approaches is generally applicable to any set of discrete levels and is not necessarily restricted to the solution of the energy eigenvalue problem, but can be applied to any observable with a discrete spectrum.

We begin with a study of the variational method, which is based upon two variation theorems, one fairly weak, and the other of which makes a stronger and more useful statement. 11

There are two general approaches commonly taken in problems of this sort.

The first, referred to as the variational method, is most commonly used to obtain information about the ground state, and low lying excited states of the system.

The second, more systematic approach is generally referred to as time-independent perturbation theory, of which there are two versions: non-degenerate perturbation theory and degenerate perturbation theory.

Either of these approaches is generally applicable to any set of discrete levels and is not necessarily restricted to the solution of the energy eigenvalue problem, but can be applied to any observable with a discrete spectrum.

We begin with a study of the variational method, which is based upon two variational theorems, one fairly weak, and the other of which makes a stronger and more useful statement. 12

We note in passing that is because of this simple variational theorem that one knows that the actual ground state energy of a many-particle system is generally lower than that of the Hartree-Fock ground state, which approximates the actual ground state in terms of a single direct product state.

Although this simple form of the variational theorem is already useful, it is possible to prove an even stronger variational statement that includes the simple bounds given above as a special case.

A statement and proof of this stronger form of the variational theorem is given in the next segment.

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The (Weak) Variational Theorem

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The Weak Variational Theorem - Let be a time-independent observable (e.g., the Hamiltonian) for a physical system having a discrete spectrum.

The normalized eigenstates of each satisfy the eigenvalue equation

.

Assume that the eigenvalues and corresponding eigenstates have been ordered, so that

and where we allow for the possibility that either the dimension of the space, or the largest eigenvalue is infinite.

15

The Weak Variational Theorem - Let be a time-independent observable (e.g., the Hamiltonian) for a physical system having a discrete spectrum.

The normalized eigenstates of each satisfy the eigenvalue equation

.

Assume that the eigenvalues and corresponding eigenstates have been ordered, so that

and where we allow for the possibility that either the dimension of the space, or the largest eigenvalue is infinite.

16

The Weak Variational Theorem - Let be a time-independent observable (e.g., the Hamiltonian) for a physical system having a discrete spectrum.

The normalized eigenstates of each satisfy the eigenvalue equation

.

Assume that the eigenvalues and corresponding eigenstates have been ordered, so that

and where we allow for the possibility that either the dimension of the space, or the largest eigenvalue is infinite.

17

The Weak Variational Theorem - Let be a time-independent observable (e.g., the Hamiltonian) for a physical system having a discrete spectrum.

The normalized eigenstates of each satisfy the eigenvalue equation

.

Assume that the eigenvalues and corresponding eigenstates have been ordered, so that

and where we allow for the possibility that either the dimension of the space, or the largest eigenvalue is infinite.

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Under these rather general circumstances, if is an arbitrary normalized state of the system it is straightforward to prove the following simple form of the variational theorem:

The mean value of with respect to an arbitrary normalized state |ψ ⟩is greater than or equal to the minimum eigenvalue and is less than or equal to the maximum eigenvalue of . , i.e., if

then

The proof follows almost trivially upon using the expansion of in its own eigenstates

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Under these rather general circumstances, if is an arbitrary normalized state of the system it is straightforward to prove the following simple form of the variational theorem:

The mean value of with respect to an arbitrary normalized state |ψ ⟩is greater than or equal to the minimum eigenvalue and is less than or equal to the maximum eigenvalue of ,

i.e., if

then

The proof follows almost trivially upon using the expansion of in its own eigenstates 20

Under these rather general circumstances, if is an arbitrary normalized state of the system it is straightforward to prove the following simple form of the variational theorem:

The mean value of with respect to an arbitrary normalized state |ψ ⟩is greater than or equal to the minimum eigenvalue and is less than or equal to the maximum eigenvalue of ,

i.e., if

then

then

The proof follows almost trivially upon using the expansion of in its own eigenstates 21

Under these rather general circumstances, if is an arbitrary normalized state of the system it is straightforward to prove the following simple form of the variational theorem:

The mean value of with respect to an arbitrary normalized state |ψ ⟩is greater than or equal to the minimum eigenvalue and is less than or equal to the maximum eigenvalue of ,

i.e., if

then

The proof follows almost trivially from the form of the expansion of

in its own eigenstates. 22

which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

so that

where we have used the assumed normalization

of the otherwise arbitrary state . 23

which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

so that

where we have used the assumed normalization

of the otherwise arbitrary state . 24

which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

so that

where we have used the assumed normalization

of the otherwise arbitrary state . 25

which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

so that

where we have used the assumed normalization

of the otherwise arbitrary state . 26

which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

so that

where we have used the assumed normalization

of the otherwise arbitrary state . 27

which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

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which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

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which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

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which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

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which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

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which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

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which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

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which we can use to express the mean value of interest in the form

We then note that each term in the last sum is itself bounded, i.e.,

So, using the fact that we find that

Thus,35

If is, in fact, the Hamiltonian of a quantum mechanical system, the variational theorem states that the ground state energy minimizes the mean value of taken with respect to the normalized states of the space.

This has interesting implications.

It means, for example, that one could simply choose a random sequence of vectors in the state space of the system and evaluate the mean value of with respect to each.

The smallest value obtained then gives an upper bound for the ground state energy of the system.

By continuing this random, or "Monte Carlo", search it is possible, in principle, to get systematically better (i.e., lower) estimates of the ground state energy and of the actual ground state itself.

36

If is, in fact, the Hamiltonian of a quantum mechanical system, the variational theorem states that the ground state energy minimizes the mean value of taken with respect to the normalized states of the space.

This has interesting implications.

It means, for example, that one could simply choose a random sequence of vectors in the state space of the system and evaluate the mean value of with respect to each.

The smallest value obtained then gives an upper bound for the ground state energy of the system.

By continuing this random, or "Monte Carlo", search it is possible, in principle, to get systematically better (i.e., lower) estimates of the ground state energy and of the actual ground state itself.

37

If is, in fact, the Hamiltonian of a quantum mechanical system, the variational theorem states that the ground state energy minimizes the mean value of taken with respect to the normalized states of the space.

This has interesting implications.

It means, for example, that one could simply choose a random sequence of vectors in the state space of the system and evaluate the mean value of with respect to each.

The smallest value obtained then gives an upper bound for the ground state energy of the system.

By continuing this random, or "Monte Carlo", search it is possible, in principle, to get systematically better (i.e., lower) estimates of the ground state energy and of the actual ground state itself.

38

If is, in fact, the Hamiltonian of a quantum mechanical system, the variational theorem states that the ground state energy minimizes the mean value of taken with respect to the normalized states of the space.

This has interesting implications.

It means, for example, that one could simply choose a random sequence of vectors in the state space of the system and evaluate the mean value of with respect to each.

The smallest value obtained then gives an upper bound for the ground state energy of the system.

By continuing this random, or "Monte Carlo", search it is possible, in principle, to get systematically better (i.e., lower) estimates of the ground state energy and of the actual ground state itself.

39

If is, in fact, the Hamiltonian of a quantum mechanical system, the variational theorem states that the ground state energy minimizes the mean value of taken with respect to the normalized states of the space.

This has interesting implications.

It means, for example, that one could simply choose a random sequence of vectors in the state space of the system and evaluate the mean value of with respect to each.

The smallest value obtained then gives an upper bound for the ground state energy of the system.

By continuing this random, or "Monte Carlo", search it is possible, in principle, to get systematically better (i.e., lower) estimates of the ground state energy and of the actual ground state itself.

40

We note in passing that it is because of this simple variational theorem that one knows that the actual ground state energy of a many-particle system is generally lower than that of the Hartree-Fock ground state, which approximates the actual ground state in terms of a single direct product state.

Although this simple form of the variational theorem is already useful, it is possible to prove an even stronger variational statement that includes the simple bounds given above as a special case.

41

We note in passing that is because of this simple variational theorem that one knows that the actual ground state energy of a many-particle system is generally lower than that of the Hartree-Fock ground state, which approximates the actual ground state in terms of a single direct product state.

Although this simple form of the variational theorem is already useful, it is possible to prove an even stronger variational statement that includes the simple bounds given above as a special case.

A statement and proof of this stronger form of the variational theorem is given in the next segment.

42

We note in passing that is because of this simple variational theorem that one knows that the actual ground state energy of a many-particle system is generally lower than that of the Hartree-Fock ground state, which approximates the actual ground state in terms of a single direct product state.

Although this simple form of the variational theorem is already useful, it is possible to prove an even stronger variational statement that includes the simple bounds given above as a special case.

A statement and proof of this stronger form of the variational theorem is given in the next segment.

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